LMFDB - Embedded newform 1452.2.b.e.725.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,2,Mod(725,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.725"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1452.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,6,0,0,0,0,0,-10,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,-32,0,6, 0,0,0,40,0,0,0,0,0,36,0,0,0,0,0,0,0,2,0,0,0,-28,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,44,0,16,0,0,0,0,0,-84,0,0,0,0,0,-58,0,0,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.5942783735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 $ x^16 - 4x^15 + 8x^14 - 12x^13 + 23x^12 - 72x^11 + 146x^10 - 176x^9 + 223x^8 - 528x^7 + 1314x^6 - 1944x^5 + 1863x^4 - 2916x^3 + 5832x^2 - 8748*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			725.3
Root			$1.63346 - 0.576037i$ of defining polynomial
Character	$\chi$	$=$	1452.725
Dual form			1452.2.b.e.725.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.05261 + 1.37550i) q^{3} -0.414160i q^{5} -1.74829i q^{7} +(-0.784027 - 2.89574i) q^{9} +2.25607i q^{13} +(0.569679 + 0.435949i) q^{15} -6.02051 q^{17} +3.55534i q^{19} +(2.40478 + 1.84027i) q^{21} +1.03009i q^{23} +4.82847 q^{25} +(4.80838 + 1.96965i) q^{27} +8.46674 q^{29} +1.08087 q^{31} -0.724072 q^{35} +7.64238 q^{37} +(-3.10324 - 2.37476i) q^{39} -2.74713 q^{41} -2.87755i q^{43} +(-1.19930 + 0.324713i) q^{45} +6.58630i q^{47} +3.94348 q^{49} +(6.33725 - 8.28124i) q^{51} -6.94627i q^{53} +(-4.89038 - 3.74238i) q^{57} +11.0168i q^{59} +11.3639i q^{61} +(-5.06259 + 1.37071i) q^{63} +0.934375 q^{65} +9.86261 q^{67} +(-1.41690 - 1.08428i) q^{69} +16.1589i q^{71} -7.18712i q^{73} +(-5.08249 + 6.64159i) q^{75} +12.8061i q^{79} +(-7.77060 + 4.54067i) q^{81} +10.5292 q^{83} +2.49345i q^{85} +(-8.91217 + 11.6460i) q^{87} +7.11103i q^{89} +3.94427 q^{91} +(-1.13773 + 1.48674i) q^{93} +1.47248 q^{95} -2.82847 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{3} - 10 q^{9} - 16 q^{15} - 32 q^{25} + 6 q^{27} + 40 q^{31} + 36 q^{37} + 2 q^{45} - 28 q^{49} + 44 q^{67} + 16 q^{69} - 84 q^{75} - 58 q^{81} - 80 q^{91} + 12 q^{93} + 64 q^{97}+O(q^{100}) $ 16 * q + 6 * q^3 - 10 * q^9 - 16 * q^15 - 32 * q^25 + 6 * q^27 + 40 * q^31 + 36 * q^37 + 2 * q^45 - 28 * q^49 + 44 * q^67 + 16 * q^69 - 84 * q^75 - 58 * q^81 - 80 * q^91 + 12 * q^93 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.05261	+	1.37550i	−0.607724	+	0.794148i
$3$	−1.05261	+	1.37550i	−0.607724	+	0.794148i
$4$		0			0
$5$		−	0.414160i		−	0.185218i	−0.995703	−	0.0926090i	$-0.970479\pi$
$5$		−	0.414160i		−	0.185218i	0.995703	−	0.0926090i	$-0.0295207\pi$
$6$		0			0
$7$		−	1.74829i		−	0.660792i	−0.943842	−	0.330396i	$-0.892818\pi$
$7$		−	1.74829i		−	0.660792i	0.943842	−	0.330396i	$-0.107182\pi$
$8$		0			0
$9$	−0.784027	−	2.89574i	−0.261342	−	0.965246i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$			2.25607i			0.625722i	0.949799	+	0.312861i	$0.101287\pi$
$13$			2.25607i			0.625722i	−0.949799	+	0.312861i	$0.898713\pi$
$14$		0			0
$15$	0.569679	+	0.435949i	0.147091	+	0.112561i
$16$		0			0
$17$	−6.02051			−1.46019			−0.730094	−	0.683347i	$-0.760524\pi$
$17$	−6.02051			−1.46019			−0.730094	+	0.683347i	$0.760524\pi$
$18$		0			0
$19$			3.55534i			0.815650i	0.913060	+	0.407825i	$0.133713\pi$
$19$			3.55534i			0.815650i	−0.913060	+	0.407825i	$0.866287\pi$
$20$		0			0
$21$	2.40478	+	1.84027i	0.524766	+	0.401579i
$22$		0			0
$23$			1.03009i			0.214789i	0.994216	+	0.107395i	$0.0342508\pi$
$23$			1.03009i			0.214789i	−0.994216	+	0.107395i	$0.965749\pi$
$24$		0			0
$25$	4.82847			0.965694
$26$		0			0
$27$	4.80838	+	1.96965i	0.925372	+	0.379059i
$28$		0			0
$29$	8.46674			1.57223			0.786117	−	0.618078i	$-0.212088\pi$
$29$	8.46674			1.57223			0.786117	+	0.618078i	$0.212088\pi$
$30$		0			0
$31$	1.08087			0.194130			0.0970649	−	0.995278i	$-0.469055\pi$
$31$	1.08087			0.194130			0.0970649	+	0.995278i	$0.469055\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.724072			−0.122391
$36$		0			0
$37$	7.64238			1.25640			0.628200	−	0.778052i	$-0.283792\pi$
$37$	7.64238			1.25640			0.628200	+	0.778052i	$0.283792\pi$
$38$		0			0
$39$	−3.10324	−	2.37476i	−0.496916	−	0.380267i
$40$		0			0
$41$	−2.74713			−0.429030			−0.214515	−	0.976721i	$-0.568817\pi$
$41$	−2.74713			−0.429030			−0.214515	+	0.976721i	$0.568817\pi$
$42$		0			0
$43$		−	2.87755i		−	0.438822i	−0.975633	−	0.219411i	$-0.929587\pi$
$43$		−	2.87755i		−	0.438822i	0.975633	−	0.219411i	$-0.0704135\pi$
$44$		0			0
$45$	−1.19930	+	0.324713i	−0.178781	+	0.0484053i
$46$		0			0
$47$			6.58630i			0.960711i	0.877074	+	0.480356i	$0.159492\pi$
$47$			6.58630i			0.960711i	−0.877074	+	0.480356i	$0.840508\pi$
$48$		0			0
$49$	3.94348			0.563354
$50$		0			0
$51$	6.33725	−	8.28124i	0.887392	−	1.15961i
$52$		0			0
$53$		−	6.94627i		−	0.954144i	−0.878864	−	0.477072i	$-0.841698\pi$
$53$		−	6.94627i		−	0.954144i	0.878864	−	0.477072i	$-0.158302\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.89038	−	3.74238i	−0.647747	−	0.495690i
$58$		0			0
$59$			11.0168i			1.43427i	0.696935	+	0.717135i	$0.254547\pi$
$59$			11.0168i			1.43427i	−0.696935	+	0.717135i	$0.745453\pi$
$60$		0			0
$61$			11.3639i			1.45500i	0.686107	+	0.727501i	$0.259318\pi$
$61$			11.3639i			1.45500i	−0.686107	+	0.727501i	$0.740682\pi$
$62$		0			0
$63$	−5.06259	+	1.37071i	−0.637827	+	0.172693i
$64$		0			0
$65$	0.934375			0.115895
$66$		0			0
$67$	9.86261			1.20491			0.602455	−	0.798153i	$-0.294189\pi$
$67$	9.86261			1.20491			0.602455	+	0.798153i	$0.294189\pi$
$68$		0			0
$69$	−1.41690	−	1.08428i	−0.170574	−	0.130533i
$70$		0			0
$71$			16.1589i			1.91771i	0.283902	+	0.958853i	$0.408371\pi$
$71$			16.1589i			1.91771i	−0.283902	+	0.958853i	$0.591629\pi$
$72$		0			0
$73$		−	7.18712i		−	0.841189i	−0.907249	−	0.420594i	$-0.861822\pi$
$73$		−	7.18712i		−	0.841189i	0.907249	−	0.420594i	$-0.138178\pi$
$74$		0			0
$75$	−5.08249	+	6.64159i	−0.586876	+	0.766904i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			12.8061i			1.44080i	0.693561	+	0.720398i	$0.256041\pi$
$79$			12.8061i			1.44080i	−0.693561	+	0.720398i	$0.743959\pi$
$80$		0			0
$81$	−7.77060	+	4.54067i	−0.863400	+	0.504519i
$82$		0			0
$83$	10.5292			1.15573			0.577863	−	0.816134i	$-0.303887\pi$
$83$	10.5292			1.15573			0.577863	+	0.816134i	$0.303887\pi$
$84$		0			0
$85$			2.49345i			0.270453i
$86$		0			0
$87$	−8.91217	+	11.6460i	−0.955485	+	1.24859i
$88$		0			0
$89$			7.11103i			0.753767i	0.926261	+	0.376884i	$0.123004\pi$
$89$			7.11103i			0.753767i	−0.926261	+	0.376884i	$0.876996\pi$
$90$		0			0
$91$	3.94427			0.413472
$92$		0			0
$93$	−1.13773	+	1.48674i	−0.117977	+	0.154168i
$94$		0			0
$95$	1.47248			0.151073
$96$		0			0
$97$	−2.82847			−0.287188			−0.143594	−	0.989637i	$-0.545866\pi$
$97$	−2.82847			−0.287188			−0.143594	+	0.989637i	$0.545866\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.5292			−1.04769			−0.523846	−	0.851813i	$-0.675503\pi$
$101$	−10.5292			−1.04769			−0.523846	+	0.851813i	$0.675503\pi$
$102$		0			0
$103$	10.7074			1.05503			0.527516	−	0.849545i	$-0.323123\pi$
$103$	10.7074			1.05503			0.527516	+	0.849545i	$0.323123\pi$
$104$		0			0
$105$	0.762165	−	0.995965i	0.0743797	−	0.0971962i
$106$		0			0
$107$	−16.7356			−1.61790			−0.808948	−	0.587880i	$-0.799963\pi$
$107$	−16.7356			−1.61790			−0.808948	+	0.587880i	$0.799963\pi$
$108$		0			0
$109$		−	19.3039i		−	1.84898i	−0.381209	−	0.924489i	$-0.624492\pi$
$109$		−	19.3039i		−	1.84898i	0.381209	−	0.924489i	$-0.375508\pi$
$110$		0			0
$111$	−8.04444	+	10.5121i	−0.763545	+	0.997768i
$112$		0			0
$113$		−	0.447653i		−	0.0421116i	−0.999778	−	0.0210558i	$-0.993297\pi$
$113$		−	0.447653i		−	0.0421116i	0.999778	−	0.0210558i	$-0.00670277\pi$
$114$		0			0
$115$	0.426623			0.0397828
$116$		0			0
$117$	6.53300	−	1.76882i	0.603976	−	0.163528i
$118$		0			0
$119$			10.5256i			0.964880i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	2.89166	−	3.77869i	0.260732	−	0.340713i
$124$		0			0
$125$		−	4.07056i		−	0.364082i
$126$		0			0
$127$		−	2.87511i		−	0.255125i	−0.991831	−	0.127562i	$-0.959285\pi$
$127$		−	2.87511i		−	0.255125i	0.991831	−	0.127562i	$-0.0407153\pi$
$128$		0			0
$129$	3.95808	+	3.02893i	0.348489	+	0.266683i
$130$		0			0
$131$	−8.65271			−0.755990			−0.377995	−	0.925808i	$-0.623386\pi$
$131$	−8.65271			−0.755990			−0.377995	+	0.925808i	$0.623386\pi$
$132$		0			0
$133$	6.21576			0.538975
$134$		0			0
$135$	0.815750	−	1.99144i	0.0702086	−	0.171396i
$136$		0			0
$137$		−	2.59281i		−	0.221519i	−0.993847	−	0.110760i	$-0.964672\pi$
$137$		−	2.59281i		−	0.221519i	0.993847	−	0.110760i	$-0.0353283\pi$
$138$		0			0
$139$			13.2962i			1.12777i	0.825855	+	0.563883i	$0.190693\pi$
$139$			13.2962i			1.12777i	−0.825855	+	0.563883i	$0.809307\pi$
$140$		0			0
$141$	−9.05949	−	6.93281i	−0.762947	−	0.583848i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	3.50658i		−	0.291206i
$146$		0			0
$147$	−4.15094	+	5.42428i	−0.342364	+	0.447387i
$148$		0			0
$149$	−7.37805			−0.604434			−0.302217	−	0.953239i	$-0.597727\pi$
$149$	−7.37805			−0.604434			−0.302217	+	0.953239i	$0.597727\pi$
$150$		0			0
$151$		−	3.17275i		−	0.258195i	−0.991632	−	0.129097i	$-0.958792\pi$
$151$		−	3.17275i		−	0.258195i	0.991632	−	0.129097i	$-0.0412080\pi$
$152$		0			0
$153$	4.72024	+	17.4338i	0.381609	+	1.40944i
$154$		0			0
$155$		−	0.447653i		−	0.0359563i
$156$		0			0
$157$	7.10075			0.566702			0.283351	−	0.959016i	$-0.408554\pi$
$157$	7.10075			0.566702			0.283351	+	0.959016i	$0.408554\pi$
$158$		0			0
$159$	9.55463	+	7.31171i	0.757731	+	0.579856i
$160$		0			0
$161$	1.80090			0.141931
$162$		0			0
$163$	15.8317			1.24004			0.620018	−	0.784588i	$-0.287125\pi$
$163$	15.8317			1.24004			0.620018	+	0.784588i	$0.287125\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.672844			−0.0520662			−0.0260331	−	0.999661i	$-0.508288\pi$
$167$	−0.672844			−0.0520662			−0.0260331	+	0.999661i	$0.508288\pi$
$168$		0			0
$169$	7.91013			0.608472
$170$		0			0
$171$	10.2953	−	2.78748i	0.787303	−	0.213164i
$172$		0			0
$173$	18.5484			1.41021			0.705105	−	0.709103i	$-0.250900\pi$
$173$	18.5484			1.41021			0.705105	+	0.709103i	$0.250900\pi$
$174$		0			0
$175$		−	8.44157i		−	0.638123i
$176$		0			0
$177$	−15.1537	−	11.5964i	−1.13902	−	0.871640i
$178$		0			0
$179$		−	8.56319i		−	0.640043i	−0.947410	−	0.320021i	$-0.896310\pi$
$179$		−	8.56319i		−	0.640043i	0.947410	−	0.320021i	$-0.103690\pi$
$180$		0			0
$181$	−12.2583			−0.911156			−0.455578	−	0.890196i	$-0.650567\pi$
$181$	−12.2583			−0.911156			−0.455578	+	0.890196i	$0.650567\pi$
$182$		0			0
$183$	−15.6311	−	11.9618i	−1.15549	−	0.884240i
$184$		0			0
$185$		−	3.16517i		−	0.232708i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	3.44352	−	8.40644i	0.250479	−	0.611478i
$190$		0			0
$191$			0.131263i			0.00949784i	0.999989	+	0.00474892i	$0.00151163\pi$
$191$			0.131263i			0.00949784i	−0.999989	+	0.00474892i	$0.998488\pi$
$192$		0			0
$193$			10.9923i			0.791241i	0.918414	+	0.395621i	$0.129470\pi$
$193$			10.9923i			0.791241i	−0.918414	+	0.395621i	$0.870530\pi$
$194$		0			0
$195$	−0.983532	+	1.28524i	−0.0704322	+	0.0920378i
$196$		0			0
$197$	13.2520			0.944164			0.472082	−	0.881555i	$-0.343503\pi$
$197$	13.2520			0.944164			0.472082	+	0.881555i	$0.343503\pi$
$198$		0			0
$199$	9.25589			0.656132			0.328066	−	0.944655i	$-0.393603\pi$
$199$	9.25589			0.656132			0.328066	+	0.944655i	$0.393603\pi$
$200$		0			0
$201$	−10.3815	+	13.5661i	−0.732253	+	0.956877i
$202$		0			0
$203$		−	14.8023i		−	1.03892i
$204$		0			0
$205$			1.13775i			0.0794640i
$206$		0			0
$207$	2.98288	−	0.807620i	0.207324	−	0.0561335i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			1.37571i			0.0947077i	0.998878	+	0.0473538i	$0.0150788\pi$
$211$			1.37571i			0.0947077i	−0.998878	+	0.0473538i	$0.984921\pi$
$212$		0			0
$213$	−22.2266	−	17.0090i	−1.52294	−	1.16544i
$214$		0			0
$215$	−1.19176			−0.0812777
$216$		0			0
$217$		−	1.88967i		−	0.128279i
$218$		0			0
$219$	9.88592	+	7.56523i	0.668028	+	0.511211i
$220$		0			0
$221$		−	13.5827i		−	0.913672i
$222$		0			0
$223$	−14.8468			−0.994213			−0.497107	−	0.867689i	$-0.665604\pi$
$223$	−14.8468			−0.994213			−0.497107	+	0.867689i	$0.665604\pi$
$224$		0			0
$225$	−3.78565	−	13.9820i	−0.252377	−	0.932133i
$226$		0			0
$227$	−4.19528			−0.278450			−0.139225	−	0.990261i	$-0.544461\pi$
$227$	−4.19528			−0.278450			−0.139225	+	0.990261i	$0.544461\pi$
$228$		0			0
$229$	3.12760			0.206678			0.103339	−	0.994646i	$-0.467047\pi$
$229$	3.12760			0.206678			0.103339	+	0.994646i	$0.467047\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	28.0718			1.83904			0.919522	−	0.393039i	$-0.128576\pi$
$233$	28.0718			1.83904			0.919522	+	0.393039i	$0.128576\pi$
$234$		0			0
$235$	2.72778			0.177941
$236$		0			0
$237$	−17.6148	−	13.4798i	−1.14421	−	0.875607i
$238$		0			0
$239$	17.2469			1.11561			0.557803	−	0.829973i	$-0.311644\pi$
$239$	17.2469			1.11561			0.557803	+	0.829973i	$0.311644\pi$
$240$		0			0
$241$		−	14.7794i		−	0.952024i	−0.879439	−	0.476012i	$-0.842082\pi$
$241$		−	14.7794i		−	0.952024i	0.879439	−	0.476012i	$-0.157918\pi$
$242$		0			0
$243$	1.93369	−	15.4681i	0.124046	−	0.992276i
$244$		0			0
$245$		−	1.63323i		−	0.104343i
$246$		0			0
$247$	−8.02110			−0.510370
$248$		0			0
$249$	−11.0831	+	14.4829i	−0.702363	+	0.917818i
$250$		0			0
$251$		−	1.55264i		−	0.0980016i	−0.998799	−	0.0490008i	$-0.984396\pi$
$251$		−	1.55264i		−	0.0980016i	0.998799	−	0.0490008i	$-0.0156037\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−3.42976	−	2.62463i	−0.214780	−	0.164361i
$256$		0			0
$257$		−	16.5858i		−	1.03460i	−0.855805	−	0.517298i	$-0.826938\pi$
$257$		−	16.5858i		−	1.03460i	0.855805	−	0.517298i	$-0.173062\pi$
$258$		0			0
$259$		−	13.3611i		−	0.830219i
$260$		0			0
$261$	−6.63815	−	24.5175i	−0.410891	−	1.51759i
$262$		0			0
$263$	−8.42736			−0.519654			−0.259827	−	0.965655i	$-0.583665\pi$
$263$	−8.42736			−0.519654			−0.259827	+	0.965655i	$0.583665\pi$
$264$		0			0
$265$	−2.87687			−0.176725
$266$		0			0
$267$	−9.78125	−	7.48513i	−0.598603	−	0.458083i
$268$		0			0
$269$		−	24.8283i		−	1.51381i	−0.653528	−	0.756903i	$-0.726712\pi$
$269$		−	24.8283i		−	1.51381i	0.653528	−	0.756903i	$-0.273288\pi$
$270$		0			0
$271$			26.8536i			1.63124i	0.578588	+	0.815620i	$0.303604\pi$
$271$			26.8536i			1.63124i	−0.578588	+	0.815620i	$0.696396\pi$
$272$		0			0
$273$	−4.15178	+	5.42537i	−0.251277	+	0.328358i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			23.4931i			1.41157i	0.708428	+	0.705783i	$0.249405\pi$
$277$			23.4931i			1.41157i	−0.708428	+	0.705783i	$0.750595\pi$
$278$		0			0
$279$	−0.847431	−	3.12991i	−0.0507343	−	0.187383i
$280$		0			0
$281$	−12.2513			−0.730853			−0.365426	−	0.930840i	$-0.619077\pi$
$281$	−12.2513			−0.730853			−0.365426	+	0.930840i	$0.619077\pi$
$282$		0			0
$283$		−	1.41202i		−	0.0839359i	−0.999119	−	0.0419680i	$-0.986637\pi$
$283$		−	1.41202i		−	0.0839359i	0.999119	−	0.0419680i	$-0.0133627\pi$
$284$		0			0
$285$	−1.54994	+	2.02540i	−0.0918108	+	0.119974i
$286$		0			0
$287$			4.80278i			0.283499i
$288$		0			0
$289$	19.2465			1.13215
$290$		0			0
$291$	2.97728	−	3.89058i	0.174531	−	0.228070i
$292$		0			0
$293$	−26.3548			−1.53966			−0.769831	−	0.638247i	$-0.779660\pi$
$293$	−26.3548			−1.53966			−0.769831	+	0.638247i	$0.779660\pi$
$294$		0			0
$295$	4.56273			0.265652
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.32396			−0.134398
$300$		0			0
$301$	−5.03079			−0.289970
$302$		0			0
$303$	11.0831	−	14.4829i	0.636708	−	0.832022i
$304$		0			0
$305$	4.70648			0.269492
$306$		0			0
$307$			6.75671i			0.385626i	0.981236	+	0.192813i	$0.0617611\pi$
$307$			6.75671i			0.385626i	−0.981236	+	0.192813i	$0.938239\pi$
$308$		0			0
$309$	−11.2707	+	14.7281i	−0.641169	+	0.837852i
$310$		0			0
$311$		−	9.59951i		−	0.544338i	−0.962249	−	0.272169i	$-0.912259\pi$
$311$		−	9.59951i		−	0.544338i	0.962249	−	0.272169i	$-0.0877410\pi$
$312$		0			0
$313$	−6.42167			−0.362974			−0.181487	−	0.983393i	$-0.558091\pi$
$313$	−6.42167			−0.362974			−0.181487	+	0.983393i	$0.558091\pi$
$314$		0			0
$315$	0.567692	+	2.09672i	0.0319858	+	0.118137i
$316$		0			0
$317$		−	13.1519i		−	0.738685i	−0.929293	−	0.369342i	$-0.879583\pi$
$317$		−	13.1519i		−	0.738685i	0.929293	−	0.369342i	$-0.120417\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	17.6161	−	23.0200i	0.983235	−	1.28485i
$322$		0			0
$323$		−	21.4049i		−	1.19100i
$324$		0			0
$325$			10.8934i			0.604256i
$326$		0			0
$327$	26.5526	+	20.3195i	1.46836	+	1.12367i
$328$		0			0
$329$	11.5148			0.634830
$330$		0			0
$331$	−4.41807			−0.242839			−0.121420	−	0.992601i	$-0.538745\pi$
$331$	−4.41807			−0.242839			−0.121420	+	0.992601i	$0.538745\pi$
$332$		0			0
$333$	−5.99184	−	22.1303i	−0.328351	−	1.21274i
$334$		0			0
$335$		−	4.08470i		−	0.223171i
$336$		0			0
$337$			12.8819i			0.701724i	0.936427	+	0.350862i	$0.114111\pi$
$337$			12.8819i			0.701724i	−0.936427	+	0.350862i	$0.885889\pi$
$338$		0			0
$339$	0.615749	+	0.471204i	0.0334429	+	0.0255923i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	19.1324i		−	1.03305i
$344$		0			0
$345$	−0.449067	+	0.586822i	−0.0241770	+	0.0315934i
$346$		0			0
$347$	27.3160			1.46640			0.733201	−	0.680012i	$-0.238025\pi$
$347$	27.3160			1.46640			0.733201	+	0.680012i	$0.238025\pi$
$348$		0			0
$349$		−	10.3120i		−	0.551991i	−0.961159	−	0.275996i	$-0.910992\pi$
$349$		−	10.3120i		−	0.551991i	0.961159	−	0.275996i	$-0.0890075\pi$
$350$		0			0
$351$	−4.44367	+	10.8481i	−0.237186	+	0.579026i
$352$		0			0
$353$		−	24.4516i		−	1.30143i	−0.759322	−	0.650715i	$-0.774469\pi$
$353$		−	24.4516i		−	1.30143i	0.759322	−	0.650715i	$-0.225531\pi$
$354$		0			0
$355$	6.69236			0.355194
$356$		0			0
$357$	−14.4780	−	11.0793i	−0.766258	−	0.586381i
$358$		0			0
$359$	−17.3566			−0.916048			−0.458024	−	0.888940i	$-0.651443\pi$
$359$	−17.3566			−0.916048			−0.458024	+	0.888940i	$0.651443\pi$
$360$		0			0
$361$	6.35959			0.334715
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.97662			−0.155803
$366$		0			0
$367$	−18.8016			−0.981437			−0.490718	−	0.871318i	$-0.663266\pi$
$367$	−18.8016			−0.981437			−0.490718	+	0.871318i	$0.663266\pi$
$368$		0			0
$369$	2.15383	+	7.95497i	0.112124	+	0.414119i
$370$		0			0
$371$	−12.1441			−0.630490
$372$		0			0
$373$			4.38788i			0.227196i	0.993527	+	0.113598i	$0.0362376\pi$
$373$			4.38788i			0.227196i	−0.993527	+	0.113598i	$0.963762\pi$
$374$		0			0
$375$	5.59907	+	4.28471i	0.289135	+	0.221261i
$376$		0			0
$377$			19.1016i			0.983781i
$378$		0			0
$379$	−15.3945			−0.790760			−0.395380	−	0.918518i	$-0.629387\pi$
$379$	−15.3945			−0.790760			−0.395380	+	0.918518i	$0.629387\pi$
$380$		0			0
$381$	3.95473	+	3.02637i	0.202607	+	0.155045i
$382$		0			0
$383$			21.6508i			1.10631i	0.833080	+	0.553153i	$0.186575\pi$
$383$			21.6508i			1.10631i	−0.833080	+	0.553153i	$0.813425\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.33262	+	2.25607i	−0.423571	+	0.114683i
$388$		0			0
$389$			9.58319i			0.485887i	0.970040	+	0.242944i	$0.0781130\pi$
$389$			9.58319i			0.485887i	−0.970040	+	0.242944i	$0.921887\pi$
$390$		0			0
$391$		−	6.20168i		−	0.313633i
$392$		0			0
$393$	9.10792	−	11.9018i	0.459434	−	0.600368i
$394$		0			0
$395$	5.30377			0.266861
$396$		0			0
$397$	4.38148			0.219900			0.109950	−	0.993937i	$-0.464931\pi$
$397$	4.38148			0.219900			0.109950	+	0.993937i	$0.464931\pi$
$398$		0			0
$399$	−6.54277	+	8.54981i	−0.327548	+	0.428026i
$400$		0			0
$401$			14.3675i			0.717477i	0.933438	+	0.358738i	$0.116793\pi$
$401$			14.3675i			0.717477i	−0.933438	+	0.358738i	$0.883207\pi$
$402$		0			0
$403$			2.43852i			0.121471i
$404$		0			0
$405$	1.88057	+	3.21827i	0.0934461	+	0.159917i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	12.8950i		−	0.637615i	−0.947820	−	0.318807i	$-0.896718\pi$
$409$		−	12.8950i		−	0.637615i	0.947820	−	0.318807i	$-0.103282\pi$
$410$		0			0
$411$	3.56643	+	2.72922i	0.175919	+	0.134623i
$412$		0			0
$413$	19.2606			0.947753
$414$		0			0
$415$		−	4.36076i		−	0.214061i
$416$		0			0
$417$	−18.2889	−	13.9957i	−0.895614	−	0.685371i
$418$		0			0
$419$			7.17498i			0.350521i	0.984522	+	0.175260i	$0.0560767\pi$
$419$			7.17498i			0.350521i	−0.984522	+	0.175260i	$0.943923\pi$
$420$		0			0
$421$	−29.3013			−1.42806			−0.714029	−	0.700116i	$-0.753131\pi$
$421$	−29.3013			−1.42806			−0.714029	+	0.700116i	$0.753131\pi$
$422$		0			0
$423$	19.0722	−	5.16384i	0.927323	−	0.251075i
$424$		0			0
$425$	−29.0699			−1.41010
$426$		0			0
$427$	19.8674			0.961453
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.0378			−0.724347			−0.362173	−	0.932111i	$-0.617965\pi$
$431$	−15.0378			−0.724347			−0.362173	+	0.932111i	$0.617965\pi$
$432$		0			0
$433$	4.39860			0.211383			0.105692	−	0.994399i	$-0.466294\pi$
$433$	4.39860			0.211383			0.105692	+	0.994399i	$0.466294\pi$
$434$		0			0
$435$	4.82332	+	3.69106i	0.231261	+	0.176973i
$436$		0			0
$437$	−3.66232			−0.175193
$438$		0			0
$439$		−	23.9994i		−	1.14543i	−0.819755	−	0.572715i	$-0.805890\pi$
$439$		−	23.9994i		−	1.14543i	0.819755	−	0.572715i	$-0.194110\pi$
$440$		0			0
$441$	−3.09180	−	11.4193i	−0.147228	−	0.543776i
$442$		0			0
$443$		−	28.4890i		−	1.35355i	−0.736188	−	0.676777i	$-0.763376\pi$
$443$		−	28.4890i		−	1.35355i	0.736188	−	0.676777i	$-0.236624\pi$
$444$		0			0
$445$	2.94510			0.139611
$446$		0			0
$447$	7.76621	−	10.1485i	0.367329	−	0.480010i
$448$		0			0
$449$			25.2944i			1.19372i	0.802346	+	0.596859i	$0.203585\pi$
$449$			25.2944i			1.19372i	−0.802346	+	0.596859i	$0.796415\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	4.36413	+	3.33967i	0.205045	+	0.156911i
$454$		0			0
$455$		−	1.63356i		−	0.0765825i
$456$		0			0
$457$			10.9704i			0.513174i	0.966521	+	0.256587i	$0.0825980\pi$
$457$			10.9704i			0.513174i	−0.966521	+	0.256587i	$0.917402\pi$
$458$		0			0
$459$	−28.9489	−	11.8583i	−1.35122	−	0.553498i
$460$		0			0
$461$	−12.5522			−0.584616			−0.292308	−	0.956324i	$-0.594423\pi$
$461$	−12.5522			−0.584616			−0.292308	+	0.956324i	$0.594423\pi$
$462$		0			0
$463$	−25.9141			−1.20433			−0.602165	−	0.798371i	$-0.705695\pi$
$463$	−25.9141			−1.20433			−0.602165	+	0.798371i	$0.705695\pi$
$464$		0			0
$465$	0.615749	+	0.471204i	0.0285547	+	0.0218515i
$466$		0			0
$467$			19.7448i			0.913679i	0.889549	+	0.456840i	$0.151019\pi$
$467$			19.7448i			0.913679i	−0.889549	+	0.456840i	$0.848981\pi$
$468$		0			0
$469$		−	17.2427i		−	0.796194i
$470$		0			0
$471$	−7.47432	+	9.76712i	−0.344398	+	0.450045i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			17.1668i			0.787669i
$476$		0			0
$477$	−20.1146	+	5.44606i	−0.920984	+	0.249358i
$478$		0			0
$479$	2.39500			0.109430			0.0547151	−	0.998502i	$-0.482575\pi$
$479$	2.39500			0.109430			0.0547151	+	0.998502i	$0.482575\pi$
$480$		0			0
$481$			17.2418i			0.786157i
$482$		0			0
$483$	−1.89565	+	2.47715i	−0.0862548	+	0.112714i
$484$		0			0
$485$			1.17144i			0.0531923i
$486$		0			0
$487$	6.00404			0.272069			0.136035	−	0.990704i	$-0.456564\pi$
$487$	6.00404			0.272069			0.136035	+	0.990704i	$0.456564\pi$
$488$		0			0
$489$	−16.6646	+	21.7766i	−0.753600	+	0.984772i
$490$		0			0
$491$	28.7885			1.29921			0.649604	−	0.760273i	$-0.274935\pi$
$491$	28.7885			1.29921			0.649604	+	0.760273i	$0.274935\pi$
$492$		0			0
$493$	−50.9741			−2.29576
$494$		0			0
$495$		0			0
$496$		0			0
$497$	28.2504			1.26720
$498$		0			0
$499$	13.3445			0.597381			0.298690	−	0.954350i	$-0.403450\pi$
$499$	13.3445			0.597381			0.298690	+	0.954350i	$0.403450\pi$
$500$		0			0
$501$	0.708242	−	0.925500i	0.0316419	−	0.0413483i
$502$		0			0
$503$	16.5891			0.739669			0.369835	−	0.929098i	$-0.379414\pi$
$503$	16.5891			0.739669			0.369835	+	0.929098i	$0.379414\pi$
$504$		0			0
$505$			4.36076i			0.194051i
$506$		0			0
$507$	−8.32628	+	10.8804i	−0.369783	+	0.483217i
$508$		0			0
$509$			7.56876i			0.335480i	0.985831	+	0.167740i	$0.0536468\pi$
$509$			7.56876i			0.335480i	−0.985831	+	0.167740i	$0.946353\pi$
$510$		0			0
$511$	−12.5652			−0.555850
$512$		0			0
$513$	−7.00276	+	17.0954i	−0.309180	+	0.754780i
$514$		0			0
$515$		−	4.43458i		−	0.195411i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−19.5242	+	25.5134i	−0.857018	+	1.11991i
$520$		0			0
$521$			24.1731i			1.05904i	0.848296	+	0.529522i	$0.177629\pi$
$521$			24.1731i			1.05904i	−0.848296	+	0.529522i	$0.822371\pi$
$522$		0			0
$523$		−	17.9282i		−	0.783945i	−0.919977	−	0.391972i	$-0.871793\pi$
$523$		−	17.9282i		−	0.783945i	0.919977	−	0.391972i	$-0.128207\pi$
$524$		0			0
$525$	11.6114	+	8.88568i	0.506764	+	0.387803i
$526$		0			0
$527$	−6.50738			−0.283466
$528$		0			0
$529$	21.9389			0.953866
$530$		0			0
$531$	31.9019	−	8.63749i	1.38442	−	0.374835i
$532$		0			0
$533$		−	6.19773i		−	0.268454i
$534$		0			0
$535$			6.93124i			0.299664i
$536$		0			0
$537$	11.7787	+	9.01369i	0.508289	+	0.388969i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			0.388767i			0.0167144i	0.999965	+	0.00835720i	$0.00266021\pi$
$541$			0.388767i			0.0167144i	−0.999965	+	0.00835720i	$0.997340\pi$
$542$		0			0
$543$	12.9032	−	16.8614i	0.553731	−	0.723592i
$544$		0			0
$545$	−7.99490			−0.342464
$546$		0			0
$547$		−	25.1091i		−	1.07359i	−0.843713	−	0.536794i	$-0.819635\pi$
$547$		−	25.1091i		−	1.07359i	0.843713	−	0.536794i	$-0.180365\pi$
$548$		0			0
$549$	32.9070	−	8.90963i	1.40443	−	0.380254i
$550$		0			0
$551$			30.1021i			1.28239i
$552$		0			0
$553$	22.3887			0.952066
$554$		0			0
$555$	4.35371	+	3.33169i	0.184805	+	0.141422i
$556$		0			0
$557$	37.3506			1.58260			0.791299	−	0.611430i	$-0.209405\pi$
$557$	37.3506			1.58260			0.791299	+	0.611430i	$0.209405\pi$
$558$		0			0
$559$	6.49195			0.274580
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.93246			0.0814434			0.0407217	−	0.999171i	$-0.487034\pi$
$563$	1.93246			0.0814434			0.0407217	+	0.999171i	$0.487034\pi$
$564$		0			0
$565$	−0.185400			−0.00779983
$566$		0			0
$567$	7.93842	+	13.5853i	0.333382	+	0.570528i
$568$		0			0
$569$	−23.9034			−1.00208			−0.501041	−	0.865423i	$-0.667050\pi$
$569$	−23.9034			−1.00208			−0.501041	+	0.865423i	$0.667050\pi$
$570$		0			0
$571$			24.0487i			1.00641i	0.864167	+	0.503204i	$0.167846\pi$
$571$			24.0487i			1.00641i	−0.864167	+	0.503204i	$0.832154\pi$
$572$		0			0
$573$	−0.180553	−	0.138168i	−0.00754269	−	0.00577207i
$574$		0			0
$575$			4.97377i			0.207421i
$576$		0			0
$577$	29.6037			1.23242			0.616209	−	0.787583i	$-0.288668\pi$
$577$	29.6037			1.23242			0.616209	+	0.787583i	$0.288668\pi$
$578$		0			0
$579$	−15.1199	−	11.5706i	−0.628363	−	0.480857i
$580$		0			0
$581$		−	18.4080i		−	0.763695i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.732576	−	2.70571i	−0.0302883	−	0.111867i
$586$		0			0
$587$			14.4688i			0.597189i	0.954380	+	0.298595i	$0.0965178\pi$
$587$			14.4688i			0.597189i	−0.954380	+	0.298595i	$0.903482\pi$
$588$		0			0
$589$			3.84285i			0.158342i
$590$		0			0
$591$	−13.9491	+	18.2281i	−0.573791	+	0.749806i
$592$		0			0
$593$	−41.4781			−1.70330			−0.851650	−	0.524111i	$-0.824398\pi$
$593$	−41.4781			−1.70330			−0.851650	+	0.524111i	$0.824398\pi$
$594$		0			0
$595$	4.35928			0.178713
$596$		0			0
$597$	−9.74283	+	12.7315i	−0.398748	+	0.521066i
$598$		0			0
$599$		−	7.19871i		−	0.294131i	−0.989127	−	0.147066i	$-0.953017\pi$
$599$		−	7.19871i		−	0.294131i	0.989127	−	0.147066i	$-0.0469829\pi$
$600$		0			0
$601$		−	30.9422i		−	1.26216i	−0.775719	−	0.631079i	$-0.782613\pi$
$601$		−	30.9422i		−	1.26216i	0.775719	−	0.631079i	$-0.217387\pi$
$602$		0			0
$603$	−7.73255	−	28.5595i	−0.314894	−	1.16303i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	44.6380i		−	1.81180i	−0.423489	−	0.905901i	$-0.639195\pi$
$607$		−	44.6380i		−	1.81180i	0.423489	−	0.905901i	$-0.360805\pi$
$608$		0			0
$609$	20.3607	+	15.5811i	0.825055	+	0.631376i
$610$		0			0
$611$	−14.8592			−0.601138
$612$		0			0
$613$			21.3685i			0.863064i	0.902098	+	0.431532i	$0.142027\pi$
$613$			21.3685i			0.863064i	−0.902098	+	0.431532i	$0.857973\pi$
$614$		0			0
$615$	−1.56498	−	1.19761i	−0.0631062	−	0.0482922i
$616$		0			0
$617$			12.7339i			0.512647i	0.966591	+	0.256324i	$0.0825113\pi$
$617$			12.7339i			0.512647i	−0.966591	+	0.256324i	$0.917489\pi$
$618$		0			0
$619$	−35.0222			−1.40766			−0.703830	−	0.710368i	$-0.748529\pi$
$619$	−35.0222			−1.40766			−0.703830	+	0.710368i	$0.748529\pi$
$620$		0			0
$621$	−2.02892	+	4.95307i	−0.0814178	+	0.198760i
$622$		0			0
$623$	12.4321			0.498083
$624$		0			0
$625$	22.4565			0.898260
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−46.0110			−1.83458
$630$		0			0
$631$	−17.0766			−0.679810			−0.339905	−	0.940460i	$-0.610395\pi$
$631$	−17.0766			−0.679810			−0.339905	+	0.940460i	$0.610395\pi$
$632$		0			0
$633$	−1.89229	−	1.44808i	−0.0752119	−	0.0575562i
$634$		0			0
$635$	−1.19076			−0.0472537
$636$		0			0
$637$			8.89678i			0.352503i
$638$		0			0
$639$	46.7919	−	12.6690i	1.85106	−	0.501178i
$640$		0			0
$641$			11.4618i			0.452713i	0.974045	+	0.226356i	$0.0726814\pi$
$641$			11.4618i			0.452713i	−0.974045	+	0.226356i	$0.927319\pi$
$642$		0			0
$643$	38.5303			1.51949			0.759744	−	0.650222i	$-0.225324\pi$
$643$	38.5303			1.51949			0.759744	+	0.650222i	$0.225324\pi$
$644$		0			0
$645$	1.25446	−	1.63928i	0.0493944	−	0.0645465i
$646$		0			0
$647$		−	20.6308i		−	0.811081i	−0.914077	−	0.405540i	$-0.867083\pi$
$647$		−	20.6308i		−	0.811081i	0.914077	−	0.405540i	$-0.132917\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	2.59926	+	1.98909i	0.101873	+	0.0779585i
$652$		0			0
$653$		−	27.7459i		−	1.08578i	−0.839804	−	0.542890i	$-0.817330\pi$
$653$		−	27.7459i		−	1.08578i	0.839804	−	0.542890i	$-0.182670\pi$
$654$		0			0
$655$			3.58361i			0.140023i
$656$		0			0
$657$	−20.8120	+	5.63489i	−0.811954	+	0.219838i
$658$		0			0
$659$	31.9195			1.24341			0.621703	−	0.783253i	$-0.286441\pi$
$659$	31.9195			1.24341			0.621703	+	0.783253i	$0.286441\pi$
$660$		0			0
$661$	−3.70222			−0.144000			−0.0719998	−	0.997405i	$-0.522938\pi$
$661$	−3.70222			−0.144000			−0.0719998	+	0.997405i	$0.522938\pi$
$662$		0			0
$663$	18.6831	+	14.2973i	0.725591	+	0.555261i
$664$		0			0
$665$		−	2.57432i		−	0.0998278i
$666$		0			0
$667$			8.72152i			0.337699i
$668$		0			0
$669$	15.6278	−	20.4218i	0.604208	−	0.789553i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			28.2008i			1.08706i	0.839390	+	0.543530i	$0.182912\pi$
$673$			28.2008i			1.08706i	−0.839390	+	0.543530i	$0.817088\pi$
$674$		0			0
$675$	23.2171	+	9.51039i	0.893627	+	0.366055i
$676$		0			0
$677$	−32.4147			−1.24580			−0.622898	−	0.782303i	$-0.714045\pi$
$677$	−32.4147			−1.24580			−0.622898	+	0.782303i	$0.714045\pi$
$678$		0			0
$679$			4.94499i			0.189771i
$680$		0			0
$681$	4.41599	−	5.77062i	0.169221	−	0.221131i
$682$		0			0
$683$		−	39.7866i		−	1.52239i	−0.648522	−	0.761196i	$-0.724613\pi$
$683$		−	39.7866i		−	1.52239i	0.648522	−	0.761196i	$-0.275387\pi$
$684$		0			0
$685$	−1.07384			−0.0410293
$686$		0			0
$687$	−3.29214	+	4.30203i	−0.125603	+	0.164133i
$688$		0			0
$689$	15.6713			0.597029
$690$		0			0
$691$	−13.5079			−0.513863			−0.256932	−	0.966430i	$-0.582712\pi$
$691$	−13.5079			−0.513863			−0.256932	+	0.966430i	$0.582712\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.50674			0.208883
$696$		0			0
$697$	16.5391			0.626464
$698$		0			0
$699$	−29.5486	+	38.6129i	−1.11763	+	1.46047i
$700$		0			0
$701$	11.4786			0.433540			0.216770	−	0.976223i	$-0.430448\pi$
$701$	11.4786			0.433540			0.216770	+	0.976223i	$0.430448\pi$
$702$		0			0
$703$			27.1712i			1.02478i
$704$		0			0
$705$	−2.87129	+	3.75208i	−0.108139	+	0.141312i
$706$		0			0
$707$			18.4080i			0.692306i
$708$		0			0
$709$	5.11097			0.191946			0.0959732	−	0.995384i	$-0.469404\pi$
$709$	5.11097			0.191946			0.0959732	+	0.995384i	$0.469404\pi$
$710$		0			0
$711$	37.0831	−	10.0403i	1.39072	−	0.376541i
$712$		0			0
$713$			1.11340i			0.0416970i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−18.1542	+	23.7231i	−0.677981	+	0.885957i
$718$		0			0
$719$			32.0243i			1.19430i	0.802128	+	0.597152i	$0.203701\pi$
$719$			32.0243i			1.19430i	−0.802128	+	0.597152i	$0.796299\pi$
$720$		0			0
$721$		−	18.7197i		−	0.697157i
$722$		0			0
$723$	20.3291	+	15.5569i	0.756048	+	0.578568i
$724$		0			0
$725$	40.8814			1.51830
$726$		0			0
$727$	−4.53747			−0.168285			−0.0841427	−	0.996454i	$-0.526815\pi$
$727$	−4.53747			−0.168285			−0.0841427	+	0.996454i	$0.526815\pi$
$728$		0			0
$729$	19.2410	+	18.9416i	0.712628	+	0.701542i
$730$		0			0
$731$			17.3243i			0.640762i
$732$		0			0
$733$		−	41.7652i		−	1.54263i	−0.636453	−	0.771316i	$-0.719599\pi$
$733$		−	41.7652i		−	1.54263i	0.636453	−	0.771316i	$-0.280401\pi$
$734$		0			0
$735$	2.24652	+	1.71916i	0.0828641	+	0.0634120i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			6.02004i			0.221451i	0.993851	+	0.110725i	$0.0353174\pi$
$739$			6.02004i			0.221451i	−0.993851	+	0.110725i	$0.964683\pi$
$740$		0			0
$741$	8.44308	−	11.0331i	0.310164	−	0.405310i
$742$		0			0
$743$	−26.0145			−0.954379			−0.477190	−	0.878800i	$-0.658345\pi$
$743$	−26.0145			−0.954379			−0.477190	+	0.878800i	$0.658345\pi$
$744$		0			0
$745$			3.05569i			0.111952i
$746$		0			0
$747$	−8.25515	−	30.4897i	−0.302040	−	1.11556i
$748$		0			0
$749$			29.2588i			1.06909i
$750$		0			0
$751$	7.52737			0.274678			0.137339	−	0.990524i	$-0.456145\pi$
$751$	7.52737			0.274678			0.137339	+	0.990524i	$0.456145\pi$
$752$		0			0
$753$	2.13566	+	1.63432i	0.0778278	+	0.0595580i
$754$		0			0
$755$	−1.31403			−0.0478223
$756$		0			0
$757$	−29.5902			−1.07548			−0.537738	−	0.843112i	$-0.680721\pi$
$757$	−29.5902			−1.07548			−0.537738	+	0.843112i	$0.680721\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.9684			0.687604			0.343802	−	0.939042i	$-0.388285\pi$
$761$	18.9684			0.687604			0.343802	+	0.939042i	$0.388285\pi$
$762$		0			0
$763$	−33.7488			−1.22179
$764$		0			0
$765$	7.22039	−	1.95494i	0.261054	−	0.0706809i
$766$		0			0
$767$	−24.8548			−0.897454
$768$		0			0
$769$			19.9245i			0.718496i	0.933242	+	0.359248i	$0.116967\pi$
$769$			19.9245i			0.718496i	−0.933242	+	0.359248i	$0.883033\pi$
$770$		0			0
$771$	22.8139	+	17.4584i	0.821622	+	0.628749i
$772$		0			0
$773$		−	32.8782i		−	1.18255i	−0.806472	−	0.591273i	$-0.798626\pi$
$773$		−	32.8782i		−	1.18255i	0.806472	−	0.591273i	$-0.201374\pi$
$774$		0			0
$775$	5.21895			0.187470
$776$		0			0
$777$	18.3783	+	14.0640i	0.659317	+	0.504544i
$778$		0			0
$779$		−	9.76698i		−	0.349938i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	40.7113	+	16.6765i	1.45490	+	0.595969i
$784$		0			0
$785$		−	2.94085i		−	0.104963i
$786$		0			0
$787$			1.03411i			0.0368621i	0.999830	+	0.0184311i	$0.00586712\pi$
$787$			1.03411i			0.0368621i	−0.999830	+	0.0184311i	$0.994133\pi$
$788$		0			0
$789$	8.87072	−	11.5919i	0.315806	−	0.412682i
$790$		0			0
$791$	−0.782627			−0.0278270
$792$		0			0
$793$	−25.6379			−0.910427
$794$		0			0
$795$	3.02822	−	3.95715i	0.107400	−	0.140345i
$796$		0			0
$797$			29.6103i			1.04885i	0.851456	+	0.524426i	$0.175720\pi$
$797$			29.6103i			1.04885i	−0.851456	+	0.524426i	$0.824280\pi$
$798$		0			0
$799$		−	39.6529i		−	1.40282i
$800$		0			0
$801$	20.5917	−	5.57524i	0.727571	−	0.196991i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	0.745861i		−	0.0262882i
$806$		0			0
$807$	34.1514	+	26.1344i	1.20219	+	0.919976i
$808$		0			0
$809$	−3.56913			−0.125484			−0.0627420	−	0.998030i	$-0.519985\pi$
$809$	−3.56913			−0.125484			−0.0627420	+	0.998030i	$0.519985\pi$
$810$		0			0
$811$			16.7897i			0.589567i	0.955564	+	0.294784i	$0.0952476\pi$
$811$			16.7897i			0.589567i	−0.955564	+	0.294784i	$0.904752\pi$
$812$		0			0
$813$	−36.9372	−	28.2663i	−1.29545	−	0.991344i
$814$		0			0
$815$		−	6.55687i		−	0.229677i
$816$		0			0
$817$	10.2306			0.357925
$818$		0			0
$819$	−3.09242	−	11.4216i	−0.108058	−	0.399102i
$820$		0			0
$821$	46.4394			1.62075			0.810373	−	0.585915i	$-0.199265\pi$
$821$	46.4394			1.62075			0.810373	+	0.585915i	$0.199265\pi$
$822$		0			0
$823$	−23.3236			−0.813010			−0.406505	−	0.913649i	$-0.633253\pi$
$823$	−23.3236			−0.813010			−0.406505	+	0.913649i	$0.633253\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.1700			−0.562287			−0.281143	−	0.959666i	$-0.590714\pi$
$827$	−16.1700			−0.562287			−0.281143	+	0.959666i	$0.590714\pi$
$828$		0			0
$829$	26.9604			0.936375			0.468187	−	0.883629i	$-0.344907\pi$
$829$	26.9604			0.936375			0.468187	+	0.883629i	$0.344907\pi$
$830$		0			0
$831$	−32.3149	−	24.7291i	−1.12099	−	0.857843i
$832$		0			0
$833$	−23.7418			−0.822603
$834$		0			0
$835$			0.278665i			0.00964360i
$836$		0			0
$837$	5.19723	+	2.12893i	0.179642	+	0.0735867i
$838$		0			0
$839$			56.2265i			1.94115i	0.240791	+	0.970577i	$0.422593\pi$
$839$			56.2265i			1.94115i	−0.240791	+	0.970577i	$0.577407\pi$
$840$		0			0
$841$	42.6856			1.47192
$842$		0			0
$843$	12.8959	−	16.8518i	0.444157	−	0.580405i
$844$		0			0
$845$		−	3.27606i		−	0.112700i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.94224	+	1.48631i	0.0666576	+	0.0510099i
$850$		0			0
$851$			7.87236i			0.269861i
$852$		0			0
$853$		−	18.2381i		−	0.624462i	−0.950006	−	0.312231i	$-0.898924\pi$
$853$		−	18.2381i		−	0.624462i	0.950006	−	0.312231i	$-0.101076\pi$
$854$		0			0
$855$	−1.15446	−	4.26391i	−0.0394818	−	0.145823i
$856$		0			0
$857$	8.99814			0.307370			0.153685	−	0.988120i	$-0.450886\pi$
$857$	8.99814			0.307370			0.153685	+	0.988120i	$0.450886\pi$
$858$		0			0
$859$	−24.5634			−0.838092			−0.419046	−	0.907965i	$-0.637635\pi$
$859$	−24.5634			−0.838092			−0.419046	+	0.907965i	$0.637635\pi$
$860$		0			0
$861$	−6.60625	−	5.05546i	−0.225140	−	0.172289i
$862$		0			0
$863$		−	8.08860i		−	0.275339i	−0.990478	−	0.137670i	$-0.956039\pi$
$863$		−	8.08860i		−	0.275339i	0.990478	−	0.137670i	$-0.0439612\pi$
$864$		0			0
$865$		−	7.68201i		−	0.261196i
$866$		0			0
$867$	−20.2591	+	26.4737i	−0.688035	+	0.899095i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			22.2508i			0.753939i
$872$		0			0
$873$	2.21760	+	8.19051i	0.0750543	+	0.277207i
$874$		0			0
$875$	−7.11652			−0.240582
$876$		0			0
$877$			1.76830i			0.0597111i	0.999554	+	0.0298556i	$0.00950473\pi$
$877$			1.76830i			0.0597111i	−0.999554	+	0.0298556i	$0.990495\pi$
$878$		0			0
$879$	27.7413	−	36.2511i	0.935690	−	1.22272i
$880$		0			0
$881$		−	5.64608i		−	0.190221i	−0.995467	−	0.0951106i	$-0.969680\pi$
$881$		−	5.64608i		−	0.190221i	0.995467	−	0.0951106i	$-0.0303205\pi$
$882$		0			0
$883$	−18.2604			−0.614511			−0.307255	−	0.951627i	$-0.599411\pi$
$883$	−18.2604			−0.614511			−0.307255	+	0.951627i	$0.599411\pi$
$884$		0			0
$885$	−4.80277	+	6.27606i	−0.161443	+	0.210967i
$886$		0			0
$887$	−19.8080			−0.665088			−0.332544	−	0.943088i	$-0.607907\pi$
$887$	−19.8080			−0.665088			−0.332544	+	0.943088i	$0.607907\pi$
$888$		0			0
$889$	−5.02653			−0.168584
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−23.4165			−0.783604
$894$		0			0
$895$	−3.54653			−0.118547
$896$		0			0
$897$	2.44623	−	3.19662i	0.0816771	−	0.106732i
$898$		0			0
$899$	9.15144			0.305217
$900$		0			0
$901$			41.8201i			1.39323i
$902$		0			0
$903$	5.29545	−	6.91987i	0.176222	−	0.230279i
$904$		0			0
$905$			5.07692i			0.168762i
$906$		0			0
$907$	−29.8058			−0.989685			−0.494843	−	0.868983i	$-0.664774\pi$
$907$	−29.8058			−0.989685			−0.494843	+	0.868983i	$0.664774\pi$
$908$		0			0
$909$	8.25515	+	30.4897i	0.273806	+	1.01128i
$910$		0			0
$911$		−	29.7037i		−	0.984128i	−0.870559	−	0.492064i	$-0.836243\pi$
$911$		−	29.7037i		−	0.984128i	0.870559	−	0.492064i	$-0.163757\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−4.95409	+	6.47379i	−0.163777	+	0.214017i
$916$		0			0
$917$			15.1274i			0.499552i
$918$		0			0
$919$			42.3723i			1.39773i	0.715251	+	0.698867i	$0.246312\pi$
$919$			42.3723i			1.39773i	−0.715251	+	0.698867i	$0.753688\pi$
$920$		0			0
$921$	−9.29389	−	7.11218i	−0.306244	−	0.234354i
$922$		0			0
$923$	−36.4556			−1.19995
$924$		0			0
$925$	36.9010			1.21330
$926$		0			0
$927$	−8.39490	−	31.0059i	−0.275725	−	1.01837i
$928$		0			0
$929$			34.8278i			1.14266i	0.820720	+	0.571331i	$0.193573\pi$
$929$			34.8278i			1.14266i	−0.820720	+	0.571331i	$0.806427\pi$
$930$		0			0
$931$			14.0204i			0.459500i
$932$		0			0
$933$	13.2042	+	10.1045i	0.432285	+	0.330808i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	50.5158i		−	1.65028i	−0.564929	−	0.825140i	$-0.691096\pi$
$937$		−	50.5158i		−	1.65028i	0.564929	−	0.825140i	$-0.308904\pi$
$938$		0			0
$939$	6.75951	−	8.83303i	0.220588	−	0.288255i
$940$		0			0
$941$	40.9400			1.33460			0.667302	−	0.744787i	$-0.267449\pi$
$941$	40.9400			1.33460			0.667302	+	0.744787i	$0.267449\pi$
$942$		0			0
$943$		−	2.82980i		−	0.0921509i
$944$		0			0
$945$	−3.48161	−	1.42617i	−0.113257	−	0.0463932i
$946$		0			0
$947$		−	2.82537i		−	0.0918122i	−0.998946	−	0.0459061i	$-0.985382\pi$
$947$		−	2.82537i		−	0.0918122i	0.998946	−	0.0459061i	$-0.0146175\pi$
$948$		0			0
$949$	16.2147			0.526350
$950$		0			0
$951$	18.0905	+	13.8438i	0.586625	+	0.448917i
$952$		0			0
$953$	−2.49808			−0.0809207			−0.0404604	−	0.999181i	$-0.512882\pi$
$953$	−2.49808			−0.0809207			−0.0404604	+	0.999181i	$0.512882\pi$
$954$		0			0
$955$	0.0543638			0.00175917
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.53299			−0.146378
$960$		0			0
$961$	−29.8317			−0.962314
$962$		0			0
$963$	13.1212	+	48.4621i	0.422825	+	1.56167i
$964$		0			0
$965$	4.55256			0.146552
$966$		0			0
$967$			16.8424i			0.541616i	0.962633	+	0.270808i	$0.0872909\pi$
$967$			16.8424i			0.541616i	−0.962633	+	0.270808i	$0.912709\pi$
$968$		0			0
$969$	29.4426	+	22.5310i	0.945832	+	0.723801i
$970$		0			0
$971$			31.7767i			1.01976i	0.860244	+	0.509882i	$0.170311\pi$
$971$			31.7767i			1.01976i	−0.860244	+	0.509882i	$0.829689\pi$
$972$		0			0
$973$	23.2456			0.745219
$974$		0			0
$975$	−14.9839	−	11.4665i	−0.479869	−	0.367221i
$976$		0			0
$977$			5.43992i			0.174038i	0.996207	+	0.0870192i	$0.0277342\pi$
$977$			5.43992i			0.174038i	−0.996207	+	0.0870192i	$0.972266\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−55.8990	+	15.1348i	−1.78472	+	0.483216i
$982$		0			0
$983$		−	39.9091i		−	1.27290i	−0.771317	−	0.636451i	$-0.780402\pi$
$983$		−	39.9091i		−	1.27290i	0.771317	−	0.636451i	$-0.219598\pi$
$984$		0			0
$985$		−	5.48843i		−	0.174876i
$986$		0			0
$987$	−12.1206	+	15.8386i	−0.385802	+	0.504149i
$988$		0			0
$989$	2.96414			0.0942541
$990$		0			0
$991$	−44.2995			−1.40722			−0.703610	−	0.710586i	$-0.748430\pi$
$991$	−44.2995			−1.40722			−0.703610	+	0.710586i	$0.748430\pi$
$992$		0			0
$993$	4.65051	−	6.07708i	0.147579	−	0.192850i
$994$		0			0
$995$		−	3.83342i		−	0.121528i
$996$		0			0
$997$		−	42.6809i		−	1.35172i	−0.737031	−	0.675859i	$-0.763773\pi$
$997$		−	42.6809i		−	1.35172i	0.737031	−	0.675859i	$-0.236227\pi$
$998$		0			0
$999$	36.7475	+	15.0528i	1.16264	+	0.476250i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.2.b.e.725.3		16
3.2	odd	2	inner	1452.2.b.e.725.1		16
11.4	even	5		132.2.p.a.17.4	yes	16
11.8	odd	10		132.2.p.a.101.3	yes	16
11.10	odd	2	inner	1452.2.b.e.725.4		16
33.8	even	10		132.2.p.a.101.4	yes	16
33.26	odd	10		132.2.p.a.17.3	✓	16
33.32	even	2	inner	1452.2.b.e.725.2		16
44.15	odd	10		528.2.bn.d.17.1		16
44.19	even	10		528.2.bn.d.497.2		16
132.59	even	10		528.2.bn.d.17.2		16
132.107	odd	10		528.2.bn.d.497.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.p.a.17.3	✓	16	33.26	odd	10
132.2.p.a.17.4	yes	16	11.4	even	5
132.2.p.a.101.3	yes	16	11.8	odd	10
132.2.p.a.101.4	yes	16	33.8	even	10
528.2.bn.d.17.1		16	44.15	odd	10
528.2.bn.d.17.2		16	132.59	even	10
528.2.bn.d.497.1		16	132.107	odd	10
528.2.bn.d.497.2		16	44.19	even	10
1452.2.b.e.725.1		16	3.2	odd	2	inner
1452.2.b.e.725.2		16	33.32	even	2	inner
1452.2.b.e.725.3		16	1.1	even	1	trivial
1452.2.b.e.725.4		16	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1452.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.05261 + 1.37550i) q^{3} -0.414160i q^{5} -1.74829i q^{7} +(-0.784027 - 2.89574i) q^{9} +2.25607i q^{13} +(0.569679 + 0.435949i) q^{15} -6.02051 q^{17} +3.55534i q^{19} +(2.40478 + 1.84027i) q^{21} +1.03009i q^{23} +4.82847 q^{25} +(4.80838 + 1.96965i) q^{27} +8.46674 q^{29} +1.08087 q^{31} -0.724072 q^{35} +7.64238 q^{37} +(-3.10324 - 2.37476i) q^{39} -2.74713 q^{41} -2.87755i q^{43} +(-1.19930 + 0.324713i) q^{45} +6.58630i q^{47} +3.94348 q^{49} +(6.33725 - 8.28124i) q^{51} -6.94627i q^{53} +(-4.89038 - 3.74238i) q^{57} +11.0168i q^{59} +11.3639i q^{61} +(-5.06259 + 1.37071i) q^{63} +0.934375 q^{65} +9.86261 q^{67} +(-1.41690 - 1.08428i) q^{69} +16.1589i q^{71} -7.18712i q^{73} +(-5.08249 + 6.64159i) q^{75} +12.8061i q^{79} +(-7.77060 + 4.54067i) q^{81} +10.5292 q^{83} +2.49345i q^{85} +(-8.91217 + 11.6460i) q^{87} +7.11103i q^{89} +3.94427 q^{91} +(-1.13773 + 1.48674i) q^{93} +1.47248 q^{95} -2.82847 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{3} - 10 q^{9} - 16 q^{15} - 32 q^{25} + 6 q^{27} + 40 q^{31} + 36 q^{37} + 2 q^{45} - 28 q^{49} + 44 q^{67} + 16 q^{69} - 84 q^{75} - 58 q^{81} - 80 q^{91} + 12 q^{93} + 64 q^{97}+O(q^{100}) \) 16 * q + 6 * q^3 - 10 * q^9 - 16 * q^15 - 32 * q^25 + 6 * q^27 + 40 * q^31 + 36 * q^37 + 2 * q^45 - 28 * q^49 + 44 * q^67 + 16 * q^69 - 84 * q^75 - 58 * q^81 - 80 * q^91 + 12 * q^93 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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